Jens Markus Melenk
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Research Interests
Jens Markus Melenk has a total of 82 co-authors. Here are the top co-authors by number of shared publications:
- Markus Faustmann
Institute for Analysis and Scientific Computing TU Wien Vienna Austria
16 shared publications - Dirk Praetorius
Institute of Analysis and Scientific Computing, TU Wien, Vienna, Austria
12 shared publications - Alexander Rieder
Institut für Analysis und Scientific Computing, TU Wien, Vienna, Austria
11 shared publications - Stefan Sauter
Institut für Mathematik, Universität Zürich, Zürich, Switzerland
10 shared publications - Barbara Wohlmuth
School of Computation, Information and Technology Department of Mathematics Technical University of Munich Boltzmannstraße 3 D-85748 Garching Germany
4 shared publications - Christoph Schwab
Seminar for Applied Mathematics, ETH Zurich, Zurich, Switzerland
4 shared publications
Latest Publications
- Open Access
- 22 September 2026
- Published by Elsevier in Computers & Mathematics with Applications
- Vol. 222, 306-343
Wavenumber-explicit analytic regularity of the heterogeneous Maxwell equations with impedance boundary conditions
- 22 September 2026
- Published by Elsevier in Computers & Mathematics with Applications
- Vol. 222, 306-343
Open AccessWavenumber-explicit analytic regularity of the heterogeneous Maxwell equations with impedance boundary conditions
- Open Access
- 12 September 2026
- Published by Springer Nature in Numerische Mathematik
- Vol. 158 (5) , 2213-2276
FEM-BEM coupling for the high-frequency Helmholtz problem
We present a wavenumber-explicit analysis of the FEM-BEM coupling methods for time-harmonic Helmholtz problems proposed in [1] for conforming discretizations and in [2] for discontinuous Galerkin (DG) volume discretizations. We show that the conditions that kh / p be sufficiently small and that $$\log (k) / p$$ be bounded imply quasi-optimality of both conforming and DG-method, where k is the wavenumber, h the mesh size, and p the approximation order. The analysis relies on a k -explicit regularity theory for a three-field coupling formulation.- 12 September 2026
- Published by Springer Nature in Numerische Mathematik
- Vol. 158 (5) , 2213-2276
Open AccessFEM-BEM coupling for the high-frequency Helmholtz problem
We present a wavenumber-explicit analysis of the FEM-BEM coupling methods for time-harmonic Helmholtz problems proposed in [1] for conforming discretizations and in [2] for discontinuous Galerkin (DG) volume discretizations. We show that the conditions that kh / p be sufficiently small and that $$\log (k) / p$$ be bounded imply quasi-optimality of both conforming and DG-method, where k is the wavenumber, h the mesh size, and p the approximation order. The analysis relies on a k -explicit regularity theory for a three-field coupling formulation. - Open Access
- 6 May 2026
- Published by Springer Nature in Zeitschrift für angewandte Mathematik und Physik
- Vol. 77 (5) , 155
Factorization of the Green’s function of high-frequency acoustic half-space impedance problems
We show that the acoustic Green’s function for a half-space impedance problem in arbitrary spatial dimension d can be written as a sum of three terms, each of which is the product of an exponential function with the eikonal in the argument and a slowly varying function. We introduce the notion of families of slowly varying functions to formulate this statement as a theorem and present its proof.
- 6 May 2026
- Published by Springer Nature in Zeitschrift für angewandte Mathematik und Physik
- Vol. 77 (5) , 155
Open AccessFactorization of the Green’s function of high-frequency acoustic half-space impedance problems
We show that the acoustic Green’s function for a half-space impedance problem in arbitrary spatial dimension d can be written as a sum of three terms, each of which is the product of an exponential function with the eikonal in the argument and a slowly varying function. We introduce the notion of families of slowly varying functions to formulate this statement as a theorem and present its proof.
- 7 February 2026
- Published by World Scientific Pub Co Pte Ltd in Analysis and Applications
- p. 1-31
Regularity of vector fields with piecewise regular curl and divergence
We consider a bounded Lipschitz domain [ ] with sufficiently smooth boundary and prove piecewise Sobolev regularity of vector fields that have piecewise regular curl and divergence, but may be discontinuous across mutually disjoint and sufficiently smooth surfaces inside of [ ]. The main idea behind our approach is to employ recently developed parametrices for the curl-operator and the regularity theory of Poisson transmission problems. We conclude our work by applying our findings to the heterogeneous time-harmonic Maxwell equations with either (a) impedance, (b) natural or (c) essential boundary conditions and provide wavenumber-explicit piecewise regularity estimates for these equations.
- 7 February 2026
- Published by World Scientific Pub Co Pte Ltd in Analysis and Applications
- p. 1-31
Regularity of vector fields with piecewise regular curl and divergence
We consider a bounded Lipschitz domain [ ] with sufficiently smooth boundary and prove piecewise Sobolev regularity of vector fields that have piecewise regular curl and divergence, but may be discontinuous across mutually disjoint and sufficiently smooth surfaces inside of [ ]. The main idea behind our approach is to employ recently developed parametrices for the curl-operator and the regularity theory of Poisson transmission problems. We conclude our work by applying our findings to the heterogeneous time-harmonic Maxwell equations with either (a) impedance, (b) natural or (c) essential boundary conditions and provide wavenumber-explicit piecewise regularity estimates for these equations.
- 19 May 2025
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Mathematical Analysis
- Vol. 57 (3) , 2559-2595
Stability Analysis for Electromagnetic Waveguides. Part 1: Acoustic and Homogeneous Electromagnetic Waveguides
In a time-harmonic setting, we show for heterogeneous acoustic and homogeneous electromagnetic wavesguides stability estimates with the stability constant depending linearly on the length [Formula: see text] of the waveguide. These stability estimates are used for the analysis of the (ideal) ultraweak (UW) variant of the discontinuous Petrov–Galerkin (DPG) method. For this UW DPG, we show that the stability deterioration with [Formula: see text] can be countered by suitably scaling the test norm of the method. We present the “full envelope approximation,” a UW DPG method based on nonpolynomial ansatz functions that allows for treating long waveguides.
- 19 May 2025
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Mathematical Analysis
- Vol. 57 (3) , 2559-2595
Stability Analysis for Electromagnetic Waveguides. Part 1: Acoustic and Homogeneous Electromagnetic Waveguides
In a time-harmonic setting, we show for heterogeneous acoustic and homogeneous electromagnetic wavesguides stability estimates with the stability constant depending linearly on the length [Formula: see text] of the waveguide. These stability estimates are used for the analysis of the (ideal) ultraweak (UW) variant of the discontinuous Petrov–Galerkin (DPG) method. For this UW DPG, we show that the stability deterioration with [Formula: see text] can be countered by suitably scaling the test norm of the method. We present the “full envelope approximation,” a UW DPG method based on nonpolynomial ansatz functions that allows for treating long waveguides.
- 11 December 2024
- Published by Institute of Mathematics, Czech Academy of Sciences in Applications of Mathematics
- Vol. 69 (5) , 1-41
A note on the shift theorem for the Laplacian in polygonal domains
Institute of Mathematics of the Czech Academy of Sciences
- 11 December 2024
- Published by Institute of Mathematics, Czech Academy of Sciences in Applications of Mathematics
- Vol. 69 (5) , 1-41
A note on the shift theorem for the Laplacian in polygonal domains
Institute of Mathematics of the Czech Academy of Sciences
- Open Access
- 6 November 2024
- Published by Elsevier in Computers & Mathematics with Applications
- Vol. 176, 324-348
An implementation of hp-FEM for the fractional Laplacian
- 6 November 2024
- Published by Elsevier in Computers & Mathematics with Applications
- Vol. 176, 324-348
Open AccessAn implementation of hp-FEM for the fractional Laplacian
- Open Access
- 12 August 2024
- Published by Elsevier in Computers & Mathematics with Applications
- Vol. 173, 1-18
Optimal convergence rates in L2 for a first order system least squares finite element method - part II: Inhomogeneous Robin boundary conditions
- 12 August 2024
- Published by Elsevier in Computers & Mathematics with Applications
- Vol. 173, 1-18
Open AccessOptimal convergence rates in L2 for a first order system least squares finite element method - part II: Inhomogeneous Robin boundary conditions
- Open Access
The Green`s function for an acoustic, half-space impedance problem Part II: Analysis of the slowly varying and the plane wave component
We show that the acoustic Green`s function for a half-space impedance problem in arbitrary spatial dimension d can be written as a sum of two terms, each of which is the product of an exponential function with the eikonal in the argument and a slowly varying function. We introduce the notion of families of slowly varying functions to formulate this statement as a theorem and present its proof.
Open AccessThe Green`s function for an acoustic, half-space impedance problem Part II: Analysis of the slowly varying and the plane wave component
We show that the acoustic Green`s function for a half-space impedance problem in arbitrary spatial dimension d can be written as a sum of two terms, each of which is the product of an exponential function with the eikonal in the argument and a slowly varying function. We introduce the notion of families of slowly varying functions to formulate this statement as a theorem and present its proof.
- Open Access
An implementation of hp-FEM for the fractional Laplacian
We consider the discretization of the $1d$-integral Dirichlet fractional Laplacian by $hp$-finite elements. We present quadrature schemes to set up the stiffness matrix and load vector that preserve the exponential convergence of $hp$-FEM on geometric meshes. The schemes are based on Gauss-Jacobi and Gauss-Legendre rules. We show that taking a number of quadrature points slightly exceeding the polynomial degree is enough to preserve root exponential convergence. The total number of algebraic operations to set up the system is $\mathcal{O}(N^{5/2})$, where $N$ is the problem size. Numerical example illustrate the analysis. We also extend our analysis to the fractional Laplacian in higher dimensions for $hp$-finite element spaces based on shape regular meshes.Open AccessAn implementation of hp-FEM for the fractional Laplacian
We consider the discretization of the $1d$-integral Dirichlet fractional Laplacian by $hp$-finite elements. We present quadrature schemes to set up the stiffness matrix and load vector that preserve the exponential convergence of $hp$-FEM on geometric meshes. The schemes are based on Gauss-Jacobi and Gauss-Legendre rules. We show that taking a number of quadrature points slightly exceeding the polynomial degree is enough to preserve root exponential convergence. The total number of algebraic operations to set up the system is $\mathcal{O}(N^{5/2})$, where $N$ is the problem size. Numerical example illustrate the analysis. We also extend our analysis to the fractional Laplacian in higher dimensions for $hp$-finite element spaces based on shape regular meshes.